Finite Group Databasebeta

Subgroups — #36475314

Order 22784.

164 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022526	C₂	1	1	yes	maximal, characteristic
2	#26022110	C₂	1	1	yes	maximal, characteristic
3	#26018453	C₂	1	1	yes	maximal, characteristic
4	#26018447	C₂	1	1	yes	maximal, characteristic
5	#26018434	C₂	1	1	yes	maximal, characteristic
6	#26018432	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1424	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
9	#19152002	—	2	1	no	—
10	#19151692	—	2	1	no	—
11	C₂×#15667918	C₂²	1	1	yes	characteristic
12	#19151424	C₂²	1	1	yes	characteristic
13	#19151423	—	2	1	no	—
14	#19151401	—	2	1	no	—
15	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
17	#366×C₈₉	C₂²	1	1	yes	characteristic
18	#366×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₅₆	—	2	1	no	—
20	#15668011	—	4	1	no	—
21	#15667977	—	4	1	no	—
22	Dic₇₁₂	—	2	1	no	—
23	Dic₇₁₂	Q₈	1	2	yes	complemented
24	C₂×#13083549	D₄	1	1	yes	characteristic
25	#15667946	—	2	1	no	—
26	#15667945	—	4	1	no	—
27	#15667935	—	4	1	no	—
28	C₈×Dic₈₉	D₄	1	1	yes	characteristic
29	#15667918	—	2	1	no	—
30	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
31	#107×C₈₉	—	2	1	no	—
32	#107×C₈₉	—	2	1	no	—
33	C₃₂×C₈₉	—	2	1	no	—
34	C₂×Dic₁₇₈	—	4	1	no	—
35	C₂×Dic₁₇₈	—	8	1	no	—
36	Dic₃₅₆	—	2	2	no	—
37	Dic₃₅₆	—	4	1	no	—
38	C₂×#10623494	D₈	1	1	yes	characteristic
39	C₄⋊Dic₈₉	—	4	1	no	—
40	#13083553	—	8	1	no	—
41	#13083553	—	8	1	no	—
42	C₄×Dic₈₉	—	8	1	no	—
43	C₄×Dic₈₉	D₈	1	1	yes	characteristic
44	#13083549	—	2	1	no	—
45	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₈₉	—	4	1	no	—
47	C₄⋊C₄×C₈₉	—	4	1	no	—
48	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₈₉	#41	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	8	1	no	—
51	C₂×C₄×C₈₉	—	8	1	no	—
52	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₈₉	—	8	1	no	—
54	C₂×Dic₈₉	—	8	1	no	—
55	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
56	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
57	Dic₁₇₈	—	4	2	no	—
58	Dic₁₇₈	—	8	1	no	—
59	Dic₁₇₈	—	16	2	no	—
60	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈×C₈₉	#137	1	1	yes	characteristic
62	#10623494	—	2	1	no	—
63	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₈₉	—	16	1	no	—
65	C₄×C₈₉	—	16	1	no	—
66	C₄×C₈₉	#474	1	1	yes	characteristic
67	C₄×C₈₉	#510	1	1	yes	characteristic
68	Dic₈₉	—	2	1	no	—
69	Dic₈₉	—	8	2	no	—
70	Dic₈₉	—	16	1	no	—
71	Dic₈₉	SD₆₄	1	2	yes	—
72	#13704	—	89	1	no	maximal, Sylow
73	C₂×C₈₉	#2224	1	1	yes	characteristic
74	C₂×C₈₉	C₂×SD₆₄	1	1	yes	characteristic
75	C₂×C₈₉	#2264	1	1	yes	characteristic
76	#2252	—	89	1	no	—
77	C₄×Q₃₂	—	89	1	no	—
78	#1424	—	89	1	no	—
79	#1424	D₈₉	1	1	yes	characteristic
80	C₄⋊C₃₂	—	89	1	no	—
81	#1416	—	89	1	no	—
82	#1416	—	89	1	no	—
83	C₈₉	#13704	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	—	89	1	no	—
85	#500	—	178	1	no	—
86	C₄×Q₁₆	—	178	1	no	—
87	C₂×C₃₂	—	89	1	no	—
88	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
89	#366	—	89	1	no	—
90	#366	—	178	1	no	—
91	#366	C₂×D₈₉	1	1	yes	characteristic
92	#366	C₂×D₈₉	1	1	yes	characteristic
93	#358	—	178	1	no	—
94	C₄×C₁₆	—	89	1	no	—
95	C₂×Q₁₆	—	178	1	no	—
96	C₄⋊Q₈	—	356	1	no	—
97	C₄×Q₈	—	356	1	no	—
98	Q₃₂	—	89	2	no	—
99	Q₃₂	—	178	1	no	—
100	C₂×C₁₆	—	89	1	no	—
101	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	2	1	no	—
104	#107	—	356	1	no	—
105	#107	—	178	1	no	—
106	#103	—	356	1	no	—
107	C₄×C₈	—	89	1	no	—
108	C₃₂	—	2	1	no	—
109	C₃₂	—	178	1	no	—
110	C₂×Q₈	—	356	1	no	—
111	C₂×Q₈	—	712	1	no	—
112	Q₁₆	—	356	1	no	—
113	Q₁₆	—	178	2	no	—
114	C₂×C₈	—	89	1	no	—
115	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	356	1	no	—
119	C₄⋊C₄	—	712	1	no	—
120	C₄⋊C₄	—	712	1	no	—
121	C₄²	—	89	1	no	—
122	C₄²	—	712	1	no	—
123	C₁₆	—	178	1	no	—
124	C₁₆	Q₈×D₈₉	1	1	yes	characteristic
125	C₁₆	#13083574	1	1	yes	characteristic
126	Q₈	—	356	2	no	—
127	Q₈	—	1424	2	no	—
128	Q₈	—	712	1	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	89	1	no	—
132	C₂×C₄	—	89	1	no	—
133	C₂×C₄	—	712	1	no	—
134	C₂×C₄	—	712	1	no	—
135	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
136	C₈	—	178	1	no	—
137	C₈	#15667990	1	1	yes	characteristic
138	C₈	#15668044	1	1	yes	characteristic
139	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
140	C₄	—	16	1	no	—
141	C₄	—	16	1	no	—
142	C₄	—	89	2	no	—
143	C₄	—	1424	1	no	—
144	C₄	—	178	1	no	—
145	C₄	—	712	2	no	—
146	C₄	#19151877	1	1	yes	characteristic
147	C₄	#19152040	1	1	yes	characteristic
148	C₂	#26022388	1	1	yes	characteristic
149	C₂	SD₆₄×D₈₉	1	1	yes	characteristic
150	C₂	#26022568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₈₉
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1424	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	#1424	D₈₉
hypocenter	C₈₉	#13704
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
23. Dic₇₁₂	Q₈	semidirect
83. C₈₉	#13704	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26022526	2
#26022110	2
#26018453	2
#26018447	2
#26018434	2
#26018432	2
C₈₉×#1424	2
#13704	89